transcendence degree

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English

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Noun

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transcendence degree (plural transcendence degrees)

  1. (algebra, field theory, of a field extension) Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K.
    • 2004, F. Hess, An Algorithm for Computing Isomorphisms of Algebraic Function Fields, Duncan Buell (editor), Algorithmic Number Theory: 6th International Symposium, ANTS-VI, LNCS 3076, page 263,
      Let and denote algebraic function fields of transcendence degree one.
    • 2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422:
      Lemma 7.19 Suppose that is a field extension[sic – meaning extension field] of transcendence degree over a field and that is not separably generated over . If are elements of such that , then for a suitable relabeling of the 's, the subfield of is of transcendence degree and is not separably generated over .
    • 2008, Bernd Sturmfels, Algorithms in Invariant Theory, 2nd edition, Springer, page 24:
      Proposition 2.1.1 Every finite matrix group has algebraically independent invariants, i.e., the ring has transcendence degree over .

Usage notes

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  • A transcendence degree is said to be of a field extension (i.e., ). More properly, it is the cardinality of a particular type of subset of the extension field , although the context of the field extension is required to make sense of the definition.
  • Relatedly, a transcendence basis of is a subset of that is algebraically independent over and such that is an algebraic extension of (that is, is an algebraic extension).
    • It can be shown that every field extension has a transcendence basis, whose cardinality, denoted or , is exactly the transcendence degree of .

Synonyms

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  • (cardinality of largest algebraically independent subset of a given extension field): transcendental degree
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Translations

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See also

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Further reading

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